Low-Latency Bootstrapping for CKKS using Roots of Unity

We introduce Sparse Roots of Unity (SPRU) bootstrapping, a new bootstrapping algorithm for the CKKS homomorphic encryption scheme for approximate arithmetic. The original CKKS bootstrapping method relies on homomorphically evaluating a polynomial that approximates modular reduction modulo q. In contrast, SPRU bootstrapping directly embeds the additive group modulo q into the complex roots of unity, which can be evaluated natively in the CKKS scheme. This approach significantly reduces the multiplicative depth required for bootstrapping, enabling the use of a smaller ring dimension and improving efficiency. In practice, using the OpenFHE C++ library, SPRU bootstrapping achieves up to a 5x reduction in latency when applied to ciphertexts with a small number of slots.